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Physics · Ch 1 — Nature of Physical World and Measurement

Dimension of Physical Quantities

1.8.1

Dimension of Physical Quantities

Every derived physical quantity can be written as some combination of the seven fundamental (base) quantities, called its dimensions, denoted with square brackets: [L][L] for length, [M][M] for mass, [T][T] for time (the three dimensions used throughout mechanics), plus [A][A] for electric current, [K][K] for temperature, [mol][\text{mol}] for amount of substance, and [cd][\text{cd}] or [Φ][\Phi] for luminous intensity.

The dimensions of a physical quantity are the powers to which the base quantities' units must be raised to represent a unit of that derived quantity.

Worked example (velocity): velocity=displacementtime=[L][T]=[M0LT−1]\text{velocity}=\dfrac{\text{displacement}}{\text{time}}=\dfrac{[L]}{[T]}=[M^0LT^{-1}] -- velocity has dimension 00 in mass, 11 in length, −1-1 in time. …

Table 1.11Dimensional Formula
Physical quantityExpressionDimensional formula
Arealength ×\times breadth[L2^2]
VolumeArea ×\times height[L3^3]
Densitymass / volume[ML−3^{-3}]
Velocitydisplacement / time[LT−1^{-1}]
Accelerationvelocity / time[LT−2^{-2}]
Momentummass ×\times velocity[MLT−1^{-1}]
Forcemass ×\times acceleration[MLT−2^{-2}]
Workforce ×\times distance[ML2^2T−2^{-2}]
Powerwork / time[ML2^2T−3^{-3}]
EnergyWork[ML2^2T−2^{-2}]
Impulseforce ×\times time[MLT−1^{-1}]
Radius of gyrationDistance[L]
Pressure (or stress)force / area[ML−1^{-1}T−2^{-2}]
Surface tensionforce / length[MT−2^{-2}]
Frequency1 / time period[T−1^{-1}]
Moment of Inertiamass ×\times (distance)2^2[ML2^2]
Moment of force (torque)force ×\times distance[ML2^2T−2^{-2}]
Angular velocityangular displacement / time[T−1^{-1}]
Angular accelerationangular velocity / time[T−2^{-2}]
Angular momentumlinear momentum ×\times distance[ML2^2T−1^{-1}]
Co-efficient of elasticitystress / strain[ML−1^{-1}T−2^{-2}]
Co-efficient of viscosity(force ×\times distance) / (area ×\times velocity)[ML−1^{-1}T−1^{-1}]
Surface energywork / area[MT−2^{-2}]
Heat capacityheat energy / temperature[ML2^2T−2^{-2}K−1^{-1}]
Chargecurrent ×\times time[AT]
Magnetic inductionforce / (current ×\times length)[MT−2^{-2}A−1^{-1}]
Force constantforce / displacement[MT−2^{-2}]
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